Noisy observations

The Extended Kalman Filter (EKF) provides an efﬁcient method for generating approximate maximum-likelihood estimates of the state of a discrete-time nonlinear dynamical system (see Chapter 1). The ﬁlter involves a recursive procedure to optimally combine noisy observations with predictions from the known dynamic model. A second use of the EKF involves estimating the parameters of a model (e.g., neural network) given clean training data of input and output data (see Chapter 2).

Stochastic control plays an important role in many scientific and applied disciplines including communications, engineering, medicine, finance and many others. It is one of the effective methods being used to find optimal decision-making strategies in applications. The book provides a collection of outstanding investigations in various aspects of stochastic systems and their behavior. The book provides a self-contained treatment on practical aspects of stochastic modeling and calculus including applications drawn from engineering, statistics, and computer science....

LEARNING STOCHASTIC NONLINEAR DYNAMICS Since the advent of cybernetics, dynamical systems have been an important modeling tool in ﬁelds ranging from engineering to the physical and social sciences. Most realistic dynamical systems models have two essential features. First, they are stochastic – the observed outputs are a noisy function of the inputs, and the dynamics itself may be driven by some unobserved noise process.

LEARNING NONLINEAR DYNAMICAL SYSTEMS USING THE EXPECTATION– MAXIMIZATION ALGORITHM
Sam Roweis and Zoubin Ghahramani
Gatsby Computational Neuroscience Unit, University College London, London U.K. (zoubin@gatsby.ucl.ac.uk)
6.1 LEARNING STOCHASTIC NONLINEAR DYNAMICS Since the advent of cybernetics, dynamical systems have been an important modeling tool in ﬁelds ranging from engineering to the physical and social sciences. Most realistic dynamical systems models have two essential features.

PLANNING AN INTERNATIONAL AUDIT: AN EMPIRICAL INVESTIGATION OF INTERNAL AUDITOR JUDGEMENT
With nearly as long a pedigree is the idea that these family background effects may
operate above the individual level. The school-level association between average student
background and average performance is typically much stronger than is the same association
at the individual level.

Much of modern digital signal processing is concernedwith the extraction of information fromsignals
whichare noisy, orwhichbehave randomlywhile still revealingsomeattributeor parameterof a system
or environment under observation. The term in popular use now for this kind of computation is
statistical signal processing, and much of this Handbook is devoted to this very subject.

In this chapter we consider a class of iterative restoration algorithms. If y is the observed noisy and
blurred signal, D the operator describing the degradation system, x the input to the system, and n
the noise added to the output signal, the input-output relation is described by [3, 51]

EXTENSIONS AND RELATED METHODS
Independent Component Analysis. Aapo Hyv¨ rinen, Juha Karhunen, Erkki Oja a Copyright  2001 John Wiley & Sons, Inc. ISBNs: 0-471-40540-X (Hardback); 0-471-22131-7 (Electronic)
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Noisy ICA
In real life, there is always some kind of noise present in the observations. Noise can correspond to actual physical noise in the measuring devices, or to inaccuracies of the model used. Therefore, it has been proposed that the independent component analysis (ICA) model should include a noise term as well.

Much of modern digital signal processing is concernedwith the extraction of information fromsignals
whichare noisy, orwhichbehave randomlywhile still revealingsomeattributeor parameterof a system
or environment under observation.

A more likely explanation is that target rate changes have been more widely anticipated
in recent years, and this squares with the Roley and Sellon (1995) observation that interest
rates rose somewhat in advance of target rate increases. Bond prices set in forward-looking
markets should respond only to the surprise element of monetary policy actions, and not
to anticipated movements in the funds rate.

Spoken dialogue managers have beneﬁted from using stochastic planners such as Markov Decision Processes (MDPs). However, so far, MDPs do not handle well noisy and ambiguous speech utterances. We use a Partially Observable Markov Decision Process (POMDP)-style approach to generate dialogue strategies by inverting the notion of dialogue state; the state represents the user’s intentions, rather than the system state.